[Encyclios]

Laws of the rolling friction of Charles-Augustin de Coulomb

1. law: the rolling friction is proportional to the normal component on the contact surface (for example in the case of horizontal surfaces we have the weight force);
2. law: rolling friction depends on the nature and state of the bodies in contact. This occurs similarly to sliding friction; an example of how this affects the friction force is that it is much easier to drive a car on asphalt than on dirt;
3. law: the rolling friction is inversely proportional to the radius of the rolling body (therefore its width): this is because the greater width of the contact surface causes a lower sinking of the body, as the unit pressure of the weight is lower; besides, the increase in the radius of the body results in an increase in the lever arm and the moment of rotation.

[Encyclios]

Rolling friction coefficient

Since the geometric manifestation of friction is the displacement of the force application point between the two members, this expression is chosen as the ratio of the displacement \(u\) to the radius of curvature of the roller \(r\):

\[\mu_r=\dfrac{u}{r}\]

which takes the name of rolling friction coefficient.

[Encyclios]

Causes of rolling friction

The characters of the contact on which the displacement of the point of application of the mutual force and therefore of the rolling friction depends, are various. The following considerations apply:

• perfect elasticity: rolling friction is null; if the materials are perfectly elastic and the shape of the bodies is perfectly regular and free of roughness, the distribution of pressures in the contact meniscus is symmetrical with respect to the normal plane of contact \(\pi_n\). Under this assumption, the line of action of the resultant of the pressures is on the symmetry plane \(\pi_n\) and \(u\) is zero;
• perfect non-elasticity: on the contrary, the material may be perfectly plastic, and then the ground level is lowered and does not rise again after passage. Most of the contact occurs at the roller’s advancing face and the \(U\) point naturally moves away from \(\pi_n\);
• elastic hysteresis: Elastic hysteresis occurs when the material is imperfectly elastic. For it, the unit tension in the material is not simply a function of the deformation, but also of the sign of its variation: for the same deformation, the tension is greater if the deformation is increasing. The distribution of pressures at contact is therefore not symmetrical, but in fact, in the front there are higher pressures than in the back;
• crushing and impact: the surface irregularities of the walls in contact give rise to loss of work:
  • by crushing of the walls that yield plastically due to the excessive concentration of the load on them;
  • by impact due to the kinematically incorrect contacts that occur in this case.
• rigid body creep: a cause of loss in imperfect rolling comes from contact creep occurring in the relative motion of the two bodies considered rigid. In this regard it should be noted that, if we consider with all rigor the motion of the individual parts taking into account their deformation, we must first define what is meant by relative motion between the two bodies, in order to have the possibility of establishing the corresponding axis of rotation;
• creep due to local deformation: Due to the deformation around the contact region, the shape and size of the surfaces change. For the two bodies the deformations of the contact areas are not the same, in fact there can be two cases:
  • the tangential action that is manifested at the contact is sufficient to prevent sliding corresponding to the different shape that the surface elements are gradually acquiring, as the stress varies according to their position along the meniscus of contact between the two bodies. In this case the work of friction for sliding is null; but it is clear that internal sliding is produced in the mass of the two bodies, the more sensitive the closer the volume element is to the contact surface. To this deformation of the volume elements corresponds work lost by elastic hysteresis;
  • the tangential action of contact is insufficient to prevent sliding at the contact; in any case it hinders it. Therefore, the phenomenon considered above is repeated to a reduced extent, with consequent loss of dynamic energy; in addition, in this case there is the loss of work corresponding to the sliding.

The resistant force \(R=F_v\) opposed by rolling friction, is the lesser the greater the radius of curvature \(r\) of the rolling body, and is calculated by the following equation:

\[F_v=\dfrac{\mu_vF_{\perp}}{r}\]

where \(\mu_v\) is the coefficient of rolling friction.